Categories of categories

TL;DR

This paper develops a generalized category theory framework within finitely complete categories equipped with a factorization system, extending concepts like finality, discreteness, and colimits beyond traditional settings.

Contribution

It introduces a broad, unified approach to category theory concepts in arbitrary finitely complete categories with factorization systems, encompassing (E,M)-category theory and (E,M)-topology.

Findings

Generalized category theory concepts expressed in a broad setting

Extension of (E,M)-category theory and (E,M)-topology

Analysis of axioms like power objects, duality, and exponentials

Abstract

A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of finality (in particular terminal objects), discreteness and components, representability, colimits and universal arrows, seem to be best expressed in this very general setting. Furthermore, at this level we are in fact doing not only (E,M)-category theory but, in a sense, also (E,M)-topology. Other axioms, regarding power objects, duality, exponentials and the arrow object, are considered.